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A104382
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Triangle, read by rows, where T(n,k) equals number of distinct partitions of triangular number n*(n+1)/2 into k different summands for n>=k>=1.
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4
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1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 12, 6, 1, 1, 10, 27, 27, 10, 1, 1, 13, 52, 84, 57, 14, 1, 1, 17, 91, 206, 221, 110, 21, 1, 1, 22, 147, 441, 674, 532, 201, 29, 1, 1, 27, 225, 864, 1747, 1945, 1175, 352, 41, 1, 1, 32, 331, 1575, 4033, 5942, 5102, 2462, 598, 55, 1, 1, 38, 469
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Secondary diagonal equals partitions of n - 1 (A000065). Third diagonal is A104384. Third column is A104385. Row sums are A104383 where limit_{n --> inf} A104383(n+1)/A104383(n) = exp(sqrt(Pi^2/6)) = 3.605822247984...
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REFERENCES
| Abramowitz, M. and Stegun, I. A. (Editors). "Partitions into Distinct Parts." S24.2.2 in Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing. New York: Dover, pp. 825-826, 1972.
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LINKS
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Eric Weisstein's World of Mathematics, Partition Function Q.
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FORMULA
| T(n, 1)=T(n, n)=1, T(n, n-1)=A000065(n-1), T(n, 2)=[(n*(n+1)/2-1)/2].
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EXAMPLE
| Rows begin:
1;
1,1;
1,2,1;
1,4,4,1;
1,7,12,6,1;
1,10,27,27,10,1;
1,13,52,84,57,14,1;
1,17,91,206,221,110,21,1;
1,22,147,441,674,532,201,29,1;
1,27,225,864,1747,1945,1175,352,41,1;
1,32,331,1575,4033,5942,5102,2462,598,55,1; ...
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PROG
| (PARI) {T(n, k)=if(n<k|k<1, 0, polcoeff(polcoeff( prod(i=1, n*(n+1)/2, 1+y*x^i, 1+x*O(x^(n*(n+1)/2))), n*(n+1)/2, x), k, y))}
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CROSSREFS
| Cf. A008289, A000009, A000065, A104383, A104384, A104385.
Sequence in context: A034368 A113582 A118245 * A086629 A203948 A156184
Adjacent sequences: A104379 A104380 A104381 * A104383 A104384 A104385
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Mar 04 2005
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